On Faces of Quasi-arithmetic Coxeter Polytopes

@article{Bogachev2020OnFO,
  title={On Faces of Quasi-arithmetic Coxeter Polytopes},
  author={Nikolay Bogachev and Alexander Kolpakov},
  journal={arXiv: Geometric Topology},
  year={2020}
}
We prove that each lower-dimensional face of a quasi-arithmetic Coxeter polytope, which happens to be itself a Coxeter polytope, is also quasi-arithmetic. We also provide a sufficient condition for a codimension $1$ face to be actually arithmetic, as well as a few computed examples. 

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References

SHOWING 1-10 OF 43 REFERENCES
On compact hyperbolic Coxeter d-polytopes with d+4 facets
We show that there is no compact hyperbolic Coxeter d-polytope with d+4 facets for d>7. This bound is sharp: examples of such polytopes up to dimension 7 were found by Bugaenko (1984). We also show
On volumes of quasi-arithmetic hyperbolic lattices
We prove that the covolume of any quasi-arithmetic hyperbolic lattice (a notion that generalizes the definition of arithmetic subgroups) is a rational multiple of the covolume of an arithmetic
On the classification of stably reflective hyperbolic Z[√2]-lattices of rank 4
  • N. Bogachev
  • Mathematics
    Доклады Академии наук
  • 2019
In this paper we prove that the fundamental polyhedron of a ℤ2-arithmetic reflection group in the three-dimensional Lobachevsky space contains an edge such that the distance between its framing faces
Non-Arithmetic Hyperbolic Reflection Groups in Higher Dimensions
We construct examples of non-arithmetic (non-cocompact) cofinite discrete reflection groups in n-dimensional Lobachevsky spaces Ln for n ≤ 18, n = 13, 15, 16, 17.
Finiteness of arithmetic Kleinian reflection groups
We prove that there are only finitely many arithmetic Kleinian maximal reflection groups. Mathematics Subject Classification (2000). Primary 30F40; Secondary 57M.
Arithmetic hyperbolic reflection groups
A hyperbolic reflection group is a discrete group generated by reflections in the faces of an $n$-dimensional hyperbolic polyhedron. This survey article is dedicated to the study of arithmetic
Hyperplane sections of polyhedra, toroidal manifolds, and discrete groups in Lobachevskii space
A bounded polyhedron is called simple, if it is the intersection of half-spaces in general position. In this paper we estimate the number and the proportion of k-dimensional faces of a simple
Finiteness of arithmetic hyperbolic reflection groups
We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.
ON ARITHMETIC GROUPS GENERATED BY REFLECTIONS IN LOBACHEVSKY SPACES
Using E. B. Vinberg's arithmeticity criterion, the author defines the notion of the Galois lattice of a discrete arithmetic group generated by reflections in a Lobachevsky space. The author proves
Infinitely many hyperbolic Coxeter groups through dimension 19
We prove the following: there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space H^n for every n < 20 (resp. n < 7). When n=7 or 8, they may be taken to
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